Dynamics of a superlattice in a dc-trigonal wave electric field

We investigate the dynamics of a superlattice with a single-band tight-binding model under a dc-trigonal wave electric field. By using Fourier analysis, we obtain analytically the conditions of the dynamics of the system: when the ratio of the Bloch frequency to electric field frequency is not an integer, it is found that the system always remains localized; when the ratio is an integer, it is found that the conditions of the localization depend on the Bloch frequency, the electric field frequency and the amplitude of the trigonal-wave electric field. The quasi-energy spectrums are also discussed, which show that the localization occurs at the collapse points of the quasi-energy band.